Subgroups — #26011841

Order 11072.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19149679	C₂	1	1	yes	maximal, characteristic
2	#19149603	C₂	1	4	yes	maximal
3	C₄×#13063436	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₄₆	—	2	1	no	—
6	C₂×#13063436	C₂²	1	2	yes	—
7	C₄⋊Dic₁₇₃	—	2	1	no	—
8	C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₇₃	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₇₃	—	2	1	no	—
11	C₄⋊C₄×C₁₇₃	—	2	1	no	—
12	C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
13	Dic₃₄₆	—	4	2	no	—
14	C₂×Dic₁₇₃	—	2	2	no	—
15	#13063436	—	2	2	no	—
16	Q₈×C₁₇₃	—	2	4	no	—
17	C₂×C₄×C₁₇₃	—	2	2	no	—
18	C₂×C₄×C₁₇₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₇₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
21	C₂²×C₁₇₃	C₂×D₄	1	1	yes	characteristic
22	C₄×C₁₇₃	—	4	2	no	—
23	C₄×C₁₇₃	SD₁₆	1	2	yes	—
24	C₄×C₁₇₃	Q₁₆	1	2	yes	—
25	C₄×C₁₇₃	#41	1	2	yes	—
26	Dic₁₇₃	—	4	2	no	—
27	C₂×C₁₇₃	#124	1	1	yes	characteristic
28	C₂×C₁₇₃	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₁₇₃	C₂×Q₁₆	1	1	yes	characteristic
30	C₁₇₃	#487	1	1	yes	characteristic, complemented, Sylow
31	#487	—	173	1	no	maximal, Sylow
32	C₄⋊Q₈	—	173	1	no	—
33	C₄⋊Q₈	D₁₇₃	1	1	yes	characteristic
34	#103	—	173	4	no	—
35	C₄×C₈	—	173	1	no	—
36	C₂×Q₈	—	346	1	no	—
37	C₂×Q₈	C₂×D₁₇₃	1	2	yes	—
38	C₂×C₈	—	173	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	—	346	1	no	—
42	C₄⋊C₄	—	173	2	no	—
43	C₄²	C₂×D₁₇₃	1	1	yes	characteristic
44	Q₈	—	2	4	no	—
45	Q₈	—	692	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	346	2	no	—
48	C₂×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
51	C₈	—	346	2	no	—
52	C₂²	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	692	2	no	—
55	C₄	#14583813	1	2	yes	—
56	C₄	#14583816	1	2	yes	—
57	C₄	#14583825	1	2	yes	—
58	C₂	#19149666	1	1	yes	characteristic
59	C₂	C₂×#14583813	1	1	yes	characteristic
60	C₂	C₂×#14583816	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₇₃
commutator, cosocle	C₂×C₄×C₁₇₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₇₃
Fitting	C₄⋊Q₈×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₇₃
hypocenter	C₁₇₃	#487
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
30. C₁₇₃	#487	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19149679	2
#19149603	2
C₄×#13063436	2
C₄⋊Q₈×C₁₇₃	2
#487	173